To solve for [tex]\((r \circ q)(3)\)[/tex] and [tex]\((q \circ r)(3)\)[/tex], we need to understand the composition of functions.
Step-by-Step Solution:
### 1. Finding [tex]\((r \circ q)(3)\)[/tex]:
The notation [tex]\((r \circ q)(3)\)[/tex] means that we first apply the function [tex]\(q\)[/tex] to [tex]\(3\)[/tex] and then apply the function [tex]\(r\)[/tex] to the result of [tex]\(q(3)\)[/tex].
#### a. Evaluate [tex]\(q(3)\)[/tex]:
Given the function [tex]\(q(x) = -2x + 1\)[/tex], we substitute [tex]\(x = 3\)[/tex]:
[tex]\[ q(3) = -2(3) + 1 \][/tex]
[tex]\[ q(3) = -6 + 1 \][/tex]
[tex]\[ q(3) = -5 \][/tex]
So, [tex]\(q(3) = -5\)[/tex].
#### b. Evaluate [tex]\(r(q(3))\)[/tex]:
Next, we substitute [tex]\(q(3)\)[/tex] into the function [tex]\(r\)[/tex]. Given [tex]\(r(x) = -x^2 - 1\)[/tex], we substitute [tex]\(x = -5\)[/tex]:
[tex]\[ r(-5) = -(-5)^2 - 1 \][/tex]
[tex]\[ r(-5) = -25 - 1 \][/tex]
[tex]\[ r(-5) = -26 \][/tex]
So, [tex]\(r(q(3)) = -26\)[/tex].
Therefore, [tex]\((r \circ q)(3) = -26\)[/tex].
### 2. Finding [tex]\((q \circ r)(3)\)[/tex]:
The notation [tex]\((q \circ r)(3)\)[/tex] means that we first apply the function [tex]\(r\)[/tex] to [tex]\(3\)[/tex] and then apply the function [tex]\(q\)[/tex] to the result of [tex]\(r(3)\)[/tex].
#### a. Evaluate [tex]\(r(3)\)[/tex]:
Given the function [tex]\(r(x) = -x^2 - 1\)[/tex], we substitute [tex]\(x = 3\)[/tex]:
[tex]\[ r(3) = -(3)^2 - 1 \][/tex]
[tex]\[ r(3) = -9 - 1 \][/tex]
[tex]\[ r(3) = -10 \][/tex]
So, [tex]\(r(3) = -10\)[/tex].
#### b. Evaluate [tex]\(q(r(3))\)[/tex]:
Next, we substitute [tex]\(r(3)\)[/tex] into the function [tex]\(q\)[/tex]. Given [tex]\(q(x) = -2x + 1\)[/tex], we substitute [tex]\(x = -10\)[/tex]:
[tex]\[ q(-10) = -2(-10) + 1 \][/tex]
[tex]\[ q(-10) = 20 + 1 \][/tex]
[tex]\[ q(-10) = 21 \][/tex]
So, [tex]\(q(r(3)) = 21\)[/tex].
Therefore, [tex]\((q \circ r)(3) = 21\)[/tex].
### Final Answers:
[tex]\[
(r \circ q)(3) = -26
\][/tex]
[tex]\[
(q \circ r)(3) = 21
\][/tex]
